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Theorem mexval 35145
Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mexval.k 𝐾 = (mTC‘𝑇)
mexval.e 𝐸 = (mEx‘𝑇)
mexval.r 𝑅 = (mREx‘𝑇)
Assertion
Ref Expression
mexval 𝐸 = (𝐾 × 𝑅)

Proof of Theorem mexval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 mexval.e . 2 𝐸 = (mEx‘𝑇)
2 fveq2 6902 . . . . . 6 (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
3 mexval.k . . . . . 6 𝐾 = (mTC‘𝑇)
42, 3eqtr4di 2786 . . . . 5 (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
5 fveq2 6902 . . . . . 6 (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
6 mexval.r . . . . . 6 𝑅 = (mREx‘𝑇)
75, 6eqtr4di 2786 . . . . 5 (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
84, 7xpeq12d 5713 . . . 4 (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅))
9 df-mex 35130 . . . 4 mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
10 fvex 6915 . . . . 5 (mTC‘𝑡) ∈ V
11 fvex 6915 . . . . 5 (mREx‘𝑡) ∈ V
1210, 11xpex 7761 . . . 4 ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V
138, 9, 12fvmpt3i 7015 . . 3 (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
14 xp0 6167 . . . . 5 (𝐾 × ∅) = ∅
1514eqcomi 2737 . . . 4 ∅ = (𝐾 × ∅)
16 fvprc 6894 . . . 4 𝑇 ∈ V → (mEx‘𝑇) = ∅)
17 fvprc 6894 . . . . . 6 𝑇 ∈ V → (mREx‘𝑇) = ∅)
186, 17eqtrid 2780 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1918xpeq2d 5712 . . . 4 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅))
2015, 16, 193eqtr4a 2794 . . 3 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
2113, 20pm2.61i 182 . 2 (mEx‘𝑇) = (𝐾 × 𝑅)
221, 21eqtri 2756 1 𝐸 = (𝐾 × 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  Vcvv 3473  c0 4326   × cxp 5680  cfv 6553  mTCcmtc 35107  mRExcmrex 35109  mExcmex 35110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-mex 35130
This theorem is referenced by:  mexval2  35146  msubff  35173  msubco  35174  msubff1  35199  mvhf  35201  msubvrs  35203
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